Torsion of digraphs and path complexes

TL;DR

This paper introduces and compares Reidemeister and analytic torsion for directed graphs using path homology, providing formulas for their computation in graph operations.

Contribution

It defines torsion concepts for digraphs via path homology and proves their equivalence, including formulas for complex graph constructions.

Findings

Reidemeister and analytic torsions are shown to be identical for digraphs.

Formulas for torsions of Cartesian products and joins of digraphs are derived.

Theoretical foundation for torsion invariants in directed graph topology.

Abstract

We define the notions of Reidemeister torsion and analytic torsion for directed graphs by means of the path homology theory introduced by the authors in \cite{Grigoryan-Lin-Muranov-Yau2013, Grigoryan-Lin-Muranov-Yau2014, Grigoryan-Lin-Muranov-Yau2015, Grigoryan-Lin-Muranov-Yau2020}. We prove the identity of the two notions of torsions as well as obtain formulas for torsions of Cartesian products and joins of digraphs.